Two Different Problems Behind One Biomarker Association
A biomarker can be difficult to interpret for two fundamentally different reasons. First, the biomarker’s underlying value may depend on latent health that also influences the event of interest. Second, the recorded biomarker may be an imperfect reading of the underlying value. The first is a problem of endogeneity; the second is a problem of measurement error. They can occur separately or together.
The distinction matters because a noisy reading does not, by itself, explain why the true biomarker is associated with unobserved event risk. Likewise, adjusting for latent health would not, by itself, make a noisy assay an exact measure. The earlier material on unobserved health as a confounder and reverse causation near an event motivates the first problem; here the focus is how it differs from imperfect measurement.
A useful schematic event model is a hazard conditional on both the biomarker and latent health:
Here, \(U_i(t)\) can affect event risk and can also be associated with \(M_i(t)\). If an analysis relates event risk to the biomarker but omits relevant latent health, the true biomarker can be endogenous in that analysis. In contrast, measurement error concerns the additional link between \(M_i(t)\) and what is observed. A common representation is \(W_i(t)=M_i(t)+\delta_i(t)\), where \(\delta_i(t)\) is measurement error.
Endogeneity: The True Biomarker Is Entangled With Latent Risk
In this setting, endogeneity is about the relationship between the true biomarker and unobserved influences on the event. For example, worsening latent health might change the biomarker while also raising event risk. An association between the true biomarker and the event can then partly reflect their shared dependence on latent health. That association is not, on its own, evidence that changing the biomarker would change event risk.
Endogeneity is not synonymous with “measured over time,” nor does it require that the biomarker be measured inaccurately. A perfectly measured biomarker can be endogenous. Reverse causation is one possible source: an approaching event may change latent health, which changes the biomarker. A shared latent cause is another. These mechanisms concern how the true biomarker relates to event risk, not whether the assay reads the biomarker correctly.
Worked Example: A True Biomarker Associated With Latent Health
Consider a hypothetical cohort with 100 people, half in a latent lower-risk health state and half in a latent higher-risk state. Suppose the biomarker is measured perfectly: the lower-risk group has \(M=10\) units and an event hazard of \(0.01\) per person-month; the higher-risk group has \(M=14\) units and a hazard of \(0.03\) per person-month. Assume, for this illustration, that latent health determines both the group’s biomarker value and its event hazard.
The difference in biomarker values is \(14-10=4\) units, and the difference in hazards is \(0.03-0.01=0.02\) events per person-month. Thus the observed group-level contrast is \(0.02/4=0.005\) additional events per person-month per biomarker unit.
This calculation describes an association, not a causal effect. In the hypothetical setup, the true biomarker is associated with latent health, and latent health determines the event hazard. The biomarker is measured without error, so measurement error cannot explain the association. Nor can the contrast tell us what would happen if the biomarker alone were changed while latent health stayed fixed.
Measurement Error: The Recorded Value Is Not the True Value
Measurement error is an observation problem. Even if the true biomarker were unrelated to latent determinants of event risk, an assay could record values that differ from the truth. A commonly used starting point is classical additive measurement error:
In the classical version, the error has mean zero and is independent of the true biomarker and other relevant quantities, conditional on measured covariates. In particular, the error does not systematically become larger for people with worse latent health or nearer to an event. These assumptions are an idealization, not a definition of every possible measurement error process.
In a linear regression, classical error in a predictor leads to a particularly clear result. Suppose \(Y=\alpha+\beta M+\varepsilon\), with \(\varepsilon\) uncorrelated with \(M\) and the measurement error \(\delta\). Also suppose \(\delta\) is uncorrelated with \(M\) and \(Y\), has mean zero, and has variance \(\sigma_\delta^2\). Regressing \(Y\) on \(W=M+\delta\) gives the following population slope:
The fraction in parentheses is the reliability ratio under these assumptions. It is between zero and one when the measurement error variance is positive, so the fitted slope is closer to zero than \(\beta\). This is called attenuation. The result is a derivation for a linear regression with a particular error structure; it is not a universal rule that every measurement error in every event-time model must attenuate an association toward zero.
Worked Example: Calculating Classical-Error Attenuation
Suppose a hypothetical linear-model population has a true biomarker variance of \(4\) squared units and a true outcome slope of \(2\) outcome units per biomarker unit. The recorded biomarker has additive, independent error with variance \(4\) squared units. Assume the error and outcome residual satisfy the conditions just stated.
The reliability ratio is
Therefore the slope from regressing the outcome on the recorded biomarker is
Under this example’s assumptions, the expected fitted slope is \(1\), rather than the true slope of \(2\). The calculation shows how measurement error alone can weaken an association. It does not establish that an event-time coefficient will be reduced by the same factor; the attenuation derivation above depends on its linear-regression assumptions.
When Both Problems Occur
In many clinical settings, latent health affects the true biomarker and event risk, while an assay adds noise to the recorded value. The biomarker is then endogenous and measured with error. It is helpful to keep the two links separate:
- Health to true marker: latent health \(U_i(t)\) is related to \(M_i(t)\), and also influences event risk.
- True marker to recorded value: \(M_i(t)\) is measured imperfectly, producing \(W_i(t)\).
The first link can make the true biomarker–event association difficult to interpret. The second means the analysis uses an imperfect proxy for that biomarker. Fixing one link does not automatically fix the other.
Worked Example: Latent Health and Assay Noise Together
Consider another hypothetical cohort in which half the people have \(U=0\) and half have \(U=1\). Let the true marker be \(M=10+4U\) units, let the event hazard be \(\lambda=0.01+0.02U\) per person-month, and let the recorded value be \(W=M+\delta\). Suppose the assay error has mean zero, variance \(4\) squared units, and is independent of \(U\) and the event process.
For \(U=0\), the true marker is \(10\) units and the hazard is \(0.01\) per person-month. For \(U=1\), the true marker is \(14\) units and the hazard is \(0.03\) per person-month. Because \(M=10+4U\), the marker and latent health have covariance
Thus the true marker is associated with latent health, which also predicts the event. That is the endogeneity issue. Separately, \(W\) differs from \(M\) because of assay error. Since \(\operatorname{Var}(M)=4^2(0.25)=4\), the reliability ratio for these population values is \(4/(4+4)=0.5\). That calculation describes the measurement component; it does not remove or quantify the confounding by \(U\). Both issues remain present in this example.
How to Tell the Problems Apart—and What to Do
No single diagnostic based only on a biomarker–event association can reliably identify which problem is present. Useful evidence depends on how the biomarker was measured and what information is available about health and the event process.
| Question | Endogeneity | Measurement error |
|---|---|---|
| What is related to what? | The true biomarker is related to unobserved event-risk determinants. | The recorded value differs from the true biomarker. |
| What information can help? | Repeated health measures, a defensible causal structure, or design information about shared causes and feedback. | Validation measurements, replicate assays, calibration studies, or known assay precision. |
| What should not be assumed? | That a biomarker association is a causal effect. | That all error is independent, additive, or guaranteed to shrink an event-model coefficient toward zero. |
Replicate assays or a validation sample can provide evidence about measurement error, though they do not by themselves resolve latent-health confounding. Conversely, measuring and modeling health-related dependence may address an endogeneity concern without correcting a noisy assay. Joint modeling can represent related longitudinal biomarker and event processes, but its usefulness depends on appropriate model assumptions and available information; merely fitting a more elaborate model does not guarantee that either source of difficulty has been identified.
Several warning signs deserve attention: biomarker readings that vary substantially across replicate assays, systematic changes in assay performance across time or health groups, and associations that change markedly after accounting for measured health history. These observations can motivate investigation, but they are not definitive tests. In particular, a single observed biomarker series generally cannot reveal how much variation is true biological change and how much is measurement error without additional assumptions or data.
Common Mistake and Key Takeaway
Wrong reasoning: “The biomarker measurement is noisy, so the biomarker–event association is explained by measurement error.” Noise may distort the association, but it does not show that the true biomarker is independent of latent health. In the other direction, finding that latent health is related to the biomarker does not show that the assay is accurate.
Correction: Ask two separate questions: (1) Is the true biomarker related to unobserved determinants of the event process? (2) How closely does the recorded value represent the true biomarker? State the assumptions used to answer each question. The answers may be “yes” to both, as in the final worked example.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A laboratory assay records a noisy value, but the true biomarker is independent of latent health and event risk. Which problem is present?
- Latent health affects both a perfectly measured biomarker and event risk. Which problem is present, and why is the biomarker association not automatically causal?
- In the linear example, what is the reliability ratio if \(\operatorname{Var}(M)=9\) and \(\sigma_\delta^2=3\)? If the true slope is \(4\), what is the expected slope using \(W\), under the stated classical-error assumptions?
- Does the classical attenuation calculation prove that measurement error always moves a time-to-event coefficient toward zero? Explain.
Solutions. (1) Measurement error is present; endogeneity is absent under the stated independence conditions. (2) Endogeneity is present because the true biomarker is related to an unobserved determinant of event risk. The association may reflect the shared dependence on latent health rather than an effect of changing the biomarker. (3) The reliability ratio is \(9/(9+3)=0.75\), so the expected slope is \(4(0.75)=3\) outcome units per biomarker unit. (4) No. The derivation assumes a linear regression, classical additive error, and the specified independence conditions. It does not establish a universal direction or magnitude of bias for event-time models.